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A congruence of generalized Bernoulli number for the character of the first kind. (English) Zbl 0982.11067
This paper deals with various congruences and their use concerning the generalized Bernoulli numbers \(B^n_\chi\), where \(\chi\) is a primitive Dirichlet character with conductor \(f\). These numbers \(B^n_\chi\) are considered to be elements of an algebraic closure \(\overline{\mathbb{Q}}_p\) of the field \(\mathbb{Q}\) of \(p\)-adic numbers (\(p\) is an odd prime).
The first congruence (Proposition 1) gives a certain \(p\)-adic approximation of \(B^n_\chi\) by means of a sum of the kind \(\sum_a\chi(a)a^n\). This congruence is then used to derive a congruence of Ferrero-Greenberg (Proposition 5) concerning the derivative \(L_p'(0,\chi\omega)\) at \(s=0\) of the Kubota-Leopoldt \(p\)-adic \(L\)-function \(L_p(s,\chi\omega)\), where \(\omega\) is the character with conductor \(p\) such that \(\omega(x)\equiv x\pmod p\) for all \(p\)-adic integers \(x\).
Using this congruence, the Ferrero-Greenberg formula [B. Ferrero and R. Greenberg, Invent. Math. 50, 91-102 (1978; Zbl 0441.12003)] is proved: if \(f\) is prime to \(p\), then \[ L_p'(0,\chi w)=(1-\chi(p))B^f_\chi\log f+\sum^{f-1}_{a=0}\chi(a)\log\Gamma\left(\frac af\right). \] \(\Gamma(x)\) is the \(p\)-adic gamma function of Morita defined by \[ \Gamma(x)=\lim_{{n\to x,r}\atop {n>0}}(-1)^n\prod^{n-1}_{a=1,(a,p)=1}a \] for each \(p\)-adic integer \(x\).

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11B68 Bernoulli and Euler numbers and polynomials
11M38 Zeta and \(L\)-functions in characteristic \(p\)